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          <h1 class="post-title" itemprop="name headline">归并排序-求逆序数算法</h1>
        

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        <p>算法：归并排序是分治法(分而治之)的一种典型应用，应用递归的思想，自顶向下思考：先假定MergeSort()可以将一个乱序的数组排好序，因此就可以开始”分”(将一个数组平均分成两部分），再”治”(分别对前后部分调用MergeSort()使它们有序)，最后再写一个合并子函数Merge()，它可以将两个有序的数组合并，Merge()实现起来比较容易．只需要管理两个指针，分别指向两个子数组的开头，开辟新内存保存中间结果，遍历完两个数组就可以完成，时间是Θ(n).</p>
<p>假定n个元素的数组调用MergeSort()需要时间T(n).因此，T(n)=2T(n/2)+Θ(n)．由主定理可知：T(n)=Θ(nlogn).归并排序算法的时间复杂度是Θ(nlogn)，空间复杂度是Θ(n).</p>
<p>由归并排序扩展可以求逆序数．</p>
<p>　　逆序数的定义：如果 i &lt; j 且A[i] &gt; A[j].则A[i]和A[j]即为逆序数对．逆序数对的个数就叫逆序数．因此求逆序数可以通过管理两个指针，两次扫描数组，蛮力法求出，显然时间复杂度是Θ(n^2).那么有更快的办法吗？答案是肯定的，利用归并排序法，稍做改进即可．在Merge()中，合并两个已经有序的数组A,B.因为A.B有序,所以,A,B各自的逆序数是0,所以AB的逆序数等于A,B之间的逆序数.</p>
<p>举个例子: A=1,4,6,7,9 B=2,3,5,10,13,21.在Merge中发现当前i号元素4比2大,那么4的逆序数需要+1,又因6,7,9都排在4后面,那么6,7,9的逆序数也应该+1,所以总体的逆序数应该加上last-i+1.(如果i号元素比B[j]小（比如4比5小）,无法确定逆序数的变化,不作任何修改).</p>
<p><img src="/2020/08/02/%E5%BD%92%E5%B9%B6%E6%8E%92%E5%BA%8F-%E6%B1%82%E9%80%86%E5%BA%8F%E6%95%B0%E7%AE%97%E6%B3%95/20130924234354406.png" alt></p>
<p>代码:</p>
<figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string">&lt;iostream&gt;</span></span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"><span class="keyword">int</span> count=<span class="number">0</span>;<span class="comment">//新增</span></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">Merge</span><span class="params">(<span class="keyword">int</span>* A,<span class="keyword">int</span> left,<span class="keyword">int</span> mid,<span class="keyword">int</span> right,<span class="keyword">int</span>* C)</span></span></span><br><span class="line"><span class="function"></span>&#123;	<span class="comment">//Merge可以将两个有序的数组排好序,时间复杂度:o(n)</span></span><br><span class="line">	<span class="keyword">int</span> i=left;</span><br><span class="line">	<span class="keyword">int</span> j=mid+<span class="number">1</span>;</span><br><span class="line">	<span class="keyword">int</span> k=left;</span><br><span class="line">	<span class="keyword">while</span>(i &lt;= mid &amp;&amp; j &lt;= right)</span><br><span class="line">	&#123;</span><br><span class="line">		<span class="keyword">if</span>(A[i] &lt;= A[j])</span><br><span class="line">			C[k++]=A[i++];</span><br><span class="line">		<span class="keyword">else</span></span><br><span class="line">		&#123;</span><br><span class="line">			C[k++]=A[j++];</span><br><span class="line">			count += mid-i+<span class="number">1</span>;<span class="comment">//新增</span></span><br><span class="line">		&#125;</span><br><span class="line">	&#125;</span><br><span class="line">	<span class="keyword">while</span>(i &lt;= mid)</span><br><span class="line">		C[k++]=A[i++];</span><br><span class="line">	<span class="keyword">while</span>(j &lt;= right)</span><br><span class="line">		C[k++]=A[j++];</span><br><span class="line">	<span class="comment">//C[]已经有序，将C[]中数据复制回原数组A[]</span></span><br><span class="line">	<span class="keyword">for</span>(<span class="keyword">int</span> i=left;i &lt;= right;++i)</span><br><span class="line">		A[i]=C[i];</span><br><span class="line">&#125;	</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">MergeSort</span><span class="params">(<span class="keyword">int</span>* A,<span class="keyword">int</span> left,<span class="keyword">int</span> right,<span class="keyword">int</span>* C)</span><span class="comment">//假定MergeSort能将一个乱序数组A排好序．</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">	<span class="keyword">if</span>(left &lt; right)</span><br><span class="line">	&#123;</span><br><span class="line">		<span class="keyword">int</span> mid=(left+right)/<span class="number">2</span>;</span><br><span class="line">		MergeSort(A,left,mid,C);<span class="comment">//排好一个数组1</span></span><br><span class="line">		MergeSort(A,mid+<span class="number">1</span>,right,C);<span class="comment">//排好一个数组2</span></span><br><span class="line">		Merge(A,left,mid,right,C); <span class="comment">//合并两个有序的数组</span></span><br><span class="line">	&#125;</span><br><span class="line">&#125;</span><br><span class="line"> </span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">main</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">	<span class="keyword">int</span> A[]=&#123;<span class="number">2</span>,<span class="number">1</span>,<span class="number">3</span>,<span class="number">6</span>,<span class="number">4</span>,<span class="number">0</span>,<span class="number">11</span>,<span class="number">3</span>,<span class="number">5</span>&#125;;</span><br><span class="line">	<span class="keyword">int</span> len=<span class="keyword">sizeof</span>(A)/<span class="keyword">sizeof</span>(A[<span class="number">0</span>]);</span><br><span class="line">	<span class="keyword">int</span> *C=<span class="keyword">new</span> <span class="keyword">int</span>[len];</span><br><span class="line">	MergeSort(A,<span class="number">0</span>,len<span class="number">-1</span>,C);</span><br><span class="line">	<span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">0</span>;i&lt;len;++i)</span><br><span class="line">		<span class="built_in">cout</span>&lt;&lt;A[i]&lt;&lt;<span class="string">' '</span>;</span><br><span class="line">	<span class="built_in">cout</span>&lt;&lt;<span class="built_in">endl</span>;</span><br><span class="line">	<span class="built_in">cout</span>&lt;&lt;count&lt;&lt;<span class="built_in">endl</span>;</span><br><span class="line">	<span class="keyword">delete</span>[] C;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>


      
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  <script type="text/javascript">
    // Popup Window;
    var isfetched = false;
    var isXml = true;
    // Search DB path;
    var search_path = "search.xml";
    if (search_path.length === 0) {
      search_path = "search.xml";
    } else if (/json$/i.test(search_path)) {
      isXml = false;
    }
    var path = "/" + search_path;
    // monitor main search box;

    var onPopupClose = function (e) {
      $('.popup').hide();
      $('#local-search-input').val('');
      $('.search-result-list').remove();
      $('#no-result').remove();
      $(".local-search-pop-overlay").remove();
      $('body').css('overflow', '');
    }

    function proceedsearch() {
      $("body")
        .append('<div class="search-popup-overlay local-search-pop-overlay"></div>')
        .css('overflow', 'hidden');
      $('.search-popup-overlay').click(onPopupClose);
      $('.popup').toggle();
      var $localSearchInput = $('#local-search-input');
      $localSearchInput.attr("autocapitalize", "none");
      $localSearchInput.attr("autocorrect", "off");
      $localSearchInput.focus();
    }

    // search function;
    var searchFunc = function(path, search_id, content_id) {
      'use strict';

      // start loading animation
      $("body")
        .append('<div class="search-popup-overlay local-search-pop-overlay">' +
          '<div id="search-loading-icon">' +
          '<i class="fa fa-spinner fa-pulse fa-5x fa-fw"></i>' +
          '</div>' +
          '</div>')
        .css('overflow', 'hidden');
      $("#search-loading-icon").css('margin', '20% auto 0 auto').css('text-align', 'center');

      $.ajax({
        url: path,
        dataType: isXml ? "xml" : "json",
        async: true,
        success: function(res) {
          // get the contents from search data
          isfetched = true;
          $('.popup').detach().appendTo('.header-inner');
          var datas = isXml ? $("entry", res).map(function() {
            return {
              title: $("title", this).text(),
              content: $("content",this).text(),
              url: $("url" , this).text()
            };
          }).get() : res;
          var input = document.getElementById(search_id);
          var resultContent = document.getElementById(content_id);
          var inputEventFunction = function() {
            var searchText = input.value.trim().toLowerCase();
            var keywords = searchText.split(/[\s\-]+/);
            if (keywords.length > 1) {
              keywords.push(searchText);
            }
            var resultItems = [];
            if (searchText.length > 0) {
              // perform local searching
              datas.forEach(function(data) {
                var isMatch = false;
                var hitCount = 0;
                var searchTextCount = 0;
                var title = data.title.trim();
                var titleInLowerCase = title.toLowerCase();
                var content = data.content.trim().replace(/<[^>]+>/g,"");
                var contentInLowerCase = content.toLowerCase();
                var articleUrl = decodeURIComponent(data.url);
                var indexOfTitle = [];
                var indexOfContent = [];
                // only match articles with not empty titles
                if(title != '') {
                  keywords.forEach(function(keyword) {
                    function getIndexByWord(word, text, caseSensitive) {
                      var wordLen = word.length;
                      if (wordLen === 0) {
                        return [];
                      }
                      var startPosition = 0, position = [], index = [];
                      if (!caseSensitive) {
                        text = text.toLowerCase();
                        word = word.toLowerCase();
                      }
                      while ((position = text.indexOf(word, startPosition)) > -1) {
                        index.push({position: position, word: word});
                        startPosition = position + wordLen;
                      }
                      return index;
                    }

                    indexOfTitle = indexOfTitle.concat(getIndexByWord(keyword, titleInLowerCase, false));
                    indexOfContent = indexOfContent.concat(getIndexByWord(keyword, contentInLowerCase, false));
                  });
                  if (indexOfTitle.length > 0 || indexOfContent.length > 0) {
                    isMatch = true;
                    hitCount = indexOfTitle.length + indexOfContent.length;
                  }
                }

                // show search results

                if (isMatch) {
                  // sort index by position of keyword

                  [indexOfTitle, indexOfContent].forEach(function (index) {
                    index.sort(function (itemLeft, itemRight) {
                      if (itemRight.position !== itemLeft.position) {
                        return itemRight.position - itemLeft.position;
                      } else {
                        return itemLeft.word.length - itemRight.word.length;
                      }
                    });
                  });

                  // merge hits into slices

                  function mergeIntoSlice(text, start, end, index) {
                    var item = index[index.length - 1];
                    var position = item.position;
                    var word = item.word;
                    var hits = [];
                    var searchTextCountInSlice = 0;
                    while (position + word.length <= end && index.length != 0) {
                      if (word === searchText) {
                        searchTextCountInSlice++;
                      }
                      hits.push({position: position, length: word.length});
                      var wordEnd = position + word.length;

                      // move to next position of hit

                      index.pop();
                      while (index.length != 0) {
                        item = index[index.length - 1];
                        position = item.position;
                        word = item.word;
                        if (wordEnd > position) {
                          index.pop();
                        } else {
                          break;
                        }
                      }
                    }
                    searchTextCount += searchTextCountInSlice;
                    return {
                      hits: hits,
                      start: start,
                      end: end,
                      searchTextCount: searchTextCountInSlice
                    };
                  }

                  var slicesOfTitle = [];
                  if (indexOfTitle.length != 0) {
                    slicesOfTitle.push(mergeIntoSlice(title, 0, title.length, indexOfTitle));
                  }

                  var slicesOfContent = [];
                  while (indexOfContent.length != 0) {
                    var item = indexOfContent[indexOfContent.length - 1];
                    var position = item.position;
                    var word = item.word;
                    // cut out 100 characters
                    var start = position - 20;
                    var end = position + 80;
                    if(start < 0){
                      start = 0;
                    }
                    if (end < position + word.length) {
                      end = position + word.length;
                    }
                    if(end > content.length){
                      end = content.length;
                    }
                    slicesOfContent.push(mergeIntoSlice(content, start, end, indexOfContent));
                  }

                  // sort slices in content by search text's count and hits' count

                  slicesOfContent.sort(function (sliceLeft, sliceRight) {
                    if (sliceLeft.searchTextCount !== sliceRight.searchTextCount) {
                      return sliceRight.searchTextCount - sliceLeft.searchTextCount;
                    } else if (sliceLeft.hits.length !== sliceRight.hits.length) {
                      return sliceRight.hits.length - sliceLeft.hits.length;
                    } else {
                      return sliceLeft.start - sliceRight.start;
                    }
                  });

                  // select top N slices in content

                  var upperBound = parseInt('1');
                  if (upperBound >= 0) {
                    slicesOfContent = slicesOfContent.slice(0, upperBound);
                  }

                  // highlight title and content

                  function highlightKeyword(text, slice) {
                    var result = '';
                    var prevEnd = slice.start;
                    slice.hits.forEach(function (hit) {
                      result += text.substring(prevEnd, hit.position);
                      var end = hit.position + hit.length;
                      result += '<b class="search-keyword">' + text.substring(hit.position, end) + '</b>';
                      prevEnd = end;
                    });
                    result += text.substring(prevEnd, slice.end);
                    return result;
                  }

                  var resultItem = '';

                  if (slicesOfTitle.length != 0) {
                    resultItem += "<li><a href='" + articleUrl + "' class='search-result-title'>" + highlightKeyword(title, slicesOfTitle[0]) + "</a>";
                  } else {
                    resultItem += "<li><a href='" + articleUrl + "' class='search-result-title'>" + title + "</a>";
                  }

                  slicesOfContent.forEach(function (slice) {
                    resultItem += "<a href='" + articleUrl + "'>" +
                      "<p class=\"search-result\">" + highlightKeyword(content, slice) +
                      "...</p>" + "</a>";
                  });

                  resultItem += "</li>";
                  resultItems.push({
                    item: resultItem,
                    searchTextCount: searchTextCount,
                    hitCount: hitCount,
                    id: resultItems.length
                  });
                }
              })
            };
            if (keywords.length === 1 && keywords[0] === "") {
              resultContent.innerHTML = '<div id="no-result"><i class="fa fa-search fa-5x" /></div>'
            } else if (resultItems.length === 0) {
              resultContent.innerHTML = '<div id="no-result"><i class="fa fa-frown-o fa-5x" /></div>'
            } else {
              resultItems.sort(function (resultLeft, resultRight) {
                if (resultLeft.searchTextCount !== resultRight.searchTextCount) {
                  return resultRight.searchTextCount - resultLeft.searchTextCount;
                } else if (resultLeft.hitCount !== resultRight.hitCount) {
                  return resultRight.hitCount - resultLeft.hitCount;
                } else {
                  return resultRight.id - resultLeft.id;
                }
              });
              var searchResultList = '<ul class=\"search-result-list\">';
              resultItems.forEach(function (result) {
                searchResultList += result.item;
              })
              searchResultList += "</ul>";
              resultContent.innerHTML = searchResultList;
            }
          }

          if ('auto' === 'auto') {
            input.addEventListener('input', inputEventFunction);
          } else {
            $('.search-icon').click(inputEventFunction);
            input.addEventListener('keypress', function (event) {
              if (event.keyCode === 13) {
                inputEventFunction();
              }
            });
          }

          // remove loading animation
          $(".local-search-pop-overlay").remove();
          $('body').css('overflow', '');

          proceedsearch();
        }
      });
    }

    // handle and trigger popup window;
    $('.popup-trigger').click(function(e) {
      e.stopPropagation();
      if (isfetched === false) {
        searchFunc(path, 'local-search-input', 'local-search-result');
      } else {
        proceedsearch();
      };
    });

    $('.popup-btn-close').click(onPopupClose);
    $('.popup').click(function(e){
      e.stopPropagation();
    });
    $(document).on('keyup', function (event) {
      var shouldDismissSearchPopup = event.which === 27 &&
        $('.search-popup').is(':visible');
      if (shouldDismissSearchPopup) {
        onPopupClose();
      }
    });
  </script>





  

  

  

  
  

  

  

  

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